In this section, we completely characterize \(q(G)\)  for all connected threshold graphs \(G\)  with trace equal to two. Note that any such graph must have at least 3 vertices. If \(G\)  is a threshold graph with trace equal to two and has 3 vertices, then \(G=K_3\) , and \(q(G)=2\) . If such a graph has 4 vertices, then either \(G \cong (0,1,0,1)\)  or \(G \cong (0,0,1,1)\) . In the later case \(G\)  is isomorphic to the complete graph on 4 vertices with one edge removed, and it is well-known that \(q(G)=2\)  in this case. For the former case, it follows that \(q(G)=3\)  from [1]}. For most of the results in this section, we assume that \(n \ge 5\) .
